Parametric Equations



Parametric Equations

The Spider and the Fly

A spider and a fly crawl so that their positions at time t (in seconds) are:

Spider: x1 = -2 + t, y1 = 5 - 2t Fly: x2 = 1 - t, y2 = 1 + t

Put your calculator in parametric and simultaneous mode and type in the parametric equations.

Window: T min = 0 X min = -6 Y min = -6

T max = 10 X max = 6 Y max = 6

T step = 1 X scl = 1 Y scl = 1

(a) At what point do their paths cross? (use the trace button to help you)

(b) Do their paths cross at the same time? (to slow up the graphing change T step to .1)

Check the table entries.

(c) Use algebra to show if the paths cross at the same time. (hint: set x1 = x2 and find t.

set y1 = y2 and find t. Do the t’s match?)

Football

From a point 90 feet directly in front of the goal posts with the crossbar 10 feet above the ground, a football is kicked at an angle of elevation [pic] with the ground and with initial velocity [pic]. If a coordinate system is set up with the ball at (0, 0) then the position (x, y) of the ball t seconds after it is kicked

is given by the parametric equations:

[pic] and [pic]

Suppose the initial velocity of the ball is 60 ft/sec

and the angle of elevation is[pic].

Put your calculator in degree, parametric and sequential modes.

To graph the goal post: x1 = 90, y1 = t

To graph the path of the ball: x2 = (60cos30)t, y2 = (60sin30)t – 16t2 (use the -0 next to x2)

Window: T min = 0 X min = 0 Y min = 0

T max = 10 X max = 100 Y max = 20

T step = 1 X scl = 1 Y scl = 1

To get a “better” picture change T step to .1

(a) Does the ball clear the crossbar on the goal posts?

(b) Keep v = 60 ft/sec and change the angle. What is the smallest angle (to the nearest whole)

that will allow the ball to pass over the crossbar?

Cool Graphs

(1) radian mode: x1 = cos(3t), y1 = sin(3t)

Window: T min = 0 X min = -1 Y min = -1

T max = 10 X max = 1 Y max = 1

T step = 1 X scl = 1 Y scl = 1

Now, try changing T step to .1

(2) x1 = cos(3t), y1 = sin(5t)

Same window

What a difference!

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(1) Spider and fly:

Spider: x1 = 3 + 2t, y1 = -2 + t Fly: x2 = -1 + 4t, y2 = 6 - 3t

Do parts (a), (b), (c)

(2) Football:

Change [pic] to 58 ft/sec

Do parts (a) and (b)

(3) If you can find a “cool” parametric graph,

give the equations and window.

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